A rambling narrative of one man's journey through mathematics

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Despite my hope to the contrary, it would appear that the math I’ve done while here so far as not parlayed into me blogging super-frequently. For what it’s worth: Life is busy. Just in case you were wondering. ^_^

Lately, I’ve been working from home more than I’ve been going to Princeton/IAS. My goal is to change that soon and I actually had a wonderful day at IAS today. I’d like to go tomorrow but I have a work meeting at the least convenient time one can imagine; there’s also no topology seminar at the University tomorrow, so I suppose I’ll be staying in and working again. No harm no foul, I suppose.

So what have I been working on? Well:

Universal Circles for Depth-One Foliations of 3-Manifolds. The gist here is: If you have a taut (e.g.) foliation on a 3-manifold, a theorem of Candel says we can find a metric on all the leaves so that they’re hyperbolic. Moreover, by tautness, you can lift to a foliation of the universal cover which is then a foliation whose leaves are hyperbolic discs. A ridiculously deep idea of Thurston was to look at the infinite circle boundaries of these disk leaves and maybe…glue them together? Canonically? And see if that gives insight about things?

You probably already know how this ends: It’s doable (because he’s Thurston) and it does provide deep insight about the downstairs manifold (see, e.g., the articles by Calegari & Dunfield and/or Fenley, or Calegari’s book…)

Now, let’s say we do this for certain classes of kind-of-understood-but-still-unknown-enough-to-be-interesting foliations like those of finite depth. Can we get cool manifold stuff by doing this process? I dunno, but maybe.

Homologies. My ATE was about Gabai’s work on foliating sutured manifolds, so studying sutured manifolds is something I’m still interested in. One way of doing that nowadays is with this colossal, ridiculously-powerful tool called Sutured Floer homology. So…you know…homology…but when talking with other grad students about the millions of homologies out there and about how nobody really understands what motivates discovers of them, I realized that there was a lot I needed to know before focusing on one homology foreverever. So I’m working on learning stuff about homologies.

Geometric Group Theory. Ian Agol is at IAS this year as the distinguished visitor and a lot of his work is on relationships between GGT and 3-manifolds. If you listen to any talk relating those two things, you realize there’s this whole dictionary of words and acronyms like QCERF and LERF and RAAG and Virtually Special, Residually Finite, etc. etc. I think in order to someday bridge the gap towards doing work like those guys do, I need to know what all these words mean, and what better time to figure that out than right now?! So yea…I’m doing that some, too.

Dirac Operators, Spin manifolds,…. At some point soon, I’m going to start working on hypercomplex geometry again, and part of that will be the study of Dirac operators. So far, there are lots of perspectives on those, so we’re going to try to first establish the explicit connections between them and then maybe…do some stuff? I dunno. I also have stuff on Clifford analysis / geometry I want to look at, as well as some more things involving generalized geometries. Lots here.

Topological Quantum Computing. This is a pipe dream until I’m able to feed my family and progress on my dissertation. It’s on the radar, though.

Okay, so this was an update! I’ve also been bookmarking some interesting proofs I’ve run across so I’ll know where to look when I decide to expand things here, and…yea.

The last time I posted something meaningful here (not counting the 2014 year-in-review and the most recent claim of attempting necromancy), it was June 2014 and I was about to embark on a summer of traveling. Around that same time, my son was 21 months old, I was working part-time at Wolfram, and I was a pre-doctoral candidate whose academic situation had gone (apparently without being blogged about) from two doctoral advisors with two separate projects to a single advisor plus a second non-advisor faculty colleague.

Typing that out makes me realize how much has changed.

For those of you keeping score, it’s now August 2015, and 13 months after the last update, lots and lots of things have changed. For example, my son is now one month away from being three years old. There’s also a lot of professional stuff, too. Let’s go somewhat chronologically.

I passed my advanced topics exam (ATE) and became a doctoral candidate. My work was on Gabai’s colossal (first) work on Reebless foliations in 3-manifolds, and while I definitely learned more significant math than I’ve ever learned, I feel like there’s so much in that paper than I’m years away from understanding.

I flew up to Baltimore to interview for an NSA gig. I didn’t get chosen.

I went to the 40th annual spring lecture series at the University of Arkansas and had a complete blast. I ended up slipping on ice, busting my ankle up pretty badly, and having some travel woes near the end but when all was said and done, I met some cool people (Benson Farb, Allen Hatcher) and saw some really great talks. Oh, and great coffee!

I went to Rhode Island College and gave an invited lecture on limit sets and computer visualization. It was an honor and I couldn’t have hoped for a better first invited lecture experience.

I finished a pretty uneventful spring semester at FSU. Lots of work. Lots and lots of work.

Once summer (2015) rolled around, I got accepted to some pretty great things:

I was fortunate enough to be awarded a pair of scholarships from the FSU math department.

And now, here we are! It’s officially September 1 (1:07am now): That means Fall semester has started at FSU (which means I’m now a fourth year doctoral student; eek) and things are back in full swing. It never gets familiar, really, no matter how many times it happens. C’est la vie, I guess.

I’ve got a bunch of stuff going on, professionally:

I’m still trying to make progress on my dissertation research (3-manifolds and, eventually, foliations).

I’m studying Dirac operators / spin manifolds / hypercomplex structures / supermanifolds / miscellaneous things that seem to get more and more into the realm of theoretical physics as we progress. This is with my non-advisor faculty colleague.

I’m trying to get a small research project going with an undergraduate at FSU on topological quantum computing (maybe Microsoft will take interest?).

Non-professionally, things have also happened. I got pretty serious into working out for a bit; later, I lost track due to travels, though I’ve since made some pretty considerable body transformations due to a healthier diet. I’ve also tuned back my Wolfram hours to give me more time to do student things; I’ve upgraded my workstations (desktop and mobile); I’ve made the switch from Windows to Linux (full-time rather than as a hobby)…

…that may actually be about it!

So there! Now we’re caught up! That means that I can pick up next time with an actual update / piece of newness / whatever. And who knows – maybe there will even be some math thrown in here! gasp

Welp, I haven’t been around these parts in a bit. That’s not to say I haven’t been working, though; I’ve just been working through things that don’t necessarily make good reading…

…like math….

Okay, so maybe that’s not entirely true. I have, however, been spending a lot of time on just a couple projects for my advisors and so there hasn’t been an abundance of topics to post about here. I think I’m getting to a point, though, where I can start posting some things here and maybe use that to reconcile the fact that I can’t understand most things I read, etc. We’ll see.

So yea…the semester’s over. It turned out not to be a terrible one for me in the end despite being pretty terrible throughout. The upshot: I managed a 4.0 that semester and ended up with advisors. That’s a victory for sure.

I’ll do my best to come back around these parts in the next day or two and post something of substance. I’d like to try to do some sort of expositing on geometric topology things (foliations, laminations, universal circles) and maybe some Clifford things too; I’d also like to attempt to reconcile my previous goal of learning how to do things from Hatcher.

So, I’ve been doing a piss-poor job of keeping this part of the internet pruned and tended to, etc. I’ve decided to stop in and give this thing a good once-over with how the semester’s been going now that the semester is (finally) nearing its end.

I gave two seminar talks at FSU’s complex analysis seminar: Complex Structures on Manifolds and Constructing Complex Manifolds Using Lie Groups. The first went pretty okay; the second was very spur of moment and came when I was in the middle of battling the flu and was unsurprisingly less-good.

I’ve had two bouts of exams so far this semester and have managed to escape both with A averages.

I recently concluded the two mandatory class-related presentations I had for the semester: I talked about Frobenius’ Theorem on the integrability of -plane distributions for my Riemannian Geometry class, and about Hyperkähler manifolds for my class on Complex Manifolds. Like above, the first of these was pretty okay and the second was kinda “meh”.

I picked doctoral advisors.

That last point is one I’m particularly happy about.

As I tend to do, I managed to pick a path that’s not the standard among students (from what I can tell) in that I picked two advisors who work in two totally unrelated fields. Be that as it may, however, I’ll officially be under the tutelage of Drs. Sergio Fenley and Craig Nolder who – respectively – study geometric topology and hypercomplex analysis/geometry. For Dr. Fenley, I’m going to be studying various aspects of foliation theory; for Dr. Nolder, I think I’m going to be studying various aspects of lots of different things.

To say I’m excited would be an understatement.

Currently, then, I’m in the process of balancing end-of-semester duties and candidacy prep duties, which means I basically haul giant stacks of books around with me 24/7 and try to read any time my eyes/brain aren’t needed for something else. It’s exhausting and nerve-wracking and brain-intensive and amazing and surreal. I literally can’t express how excited I am.

When classes start back on Monday, there will be one week of non-finals classes followed by one week of finals; over the course of those two weeks, I’ll have lots of TAing to do and lots of exams to take. When those weeks are over, though, I’ll be enveloping myself in reading roughly 20 hours a day.

Or thereabouts.

I think that’s about all I’ve got presently. I’ve been on the look-out for various fellowship/scholarship opportunities, as well as various summer programs and internships, etc. I’ll try to post progress on those fronts (and others, too) here as I remember. Between all that, I think it’s safe to say that my updating of Hatcher solutions is on the (very very far) back burner for a bit, but if I’m able, I plan to spend time going through, correcting the screw-ups that exist (believe me, there are many) and trying to get generally better-familiarized with the techniques necessary to master that material.

I just wanted to drop in and update here. I haven’t been posting much in the last day or two, but not because I haven’t been workin’ it!

Here’s what’s been going on.

Wednesday, I stayed home and had a Clifford Analysis day. I read a solid three or four pages of my professor’s paper before calling it a day.

Because I felt like I hadn’t done enough on the Clifford front, I went to my office Thursday armed with new writing supplies and spent a solid few hours verifying the claims made in the aforementioned three or four pages I’d read. That was a good feeling.

Friday was (differential) geometry day, and I started the day working some “trivial” problems from Spivak’s little book. In the middle of the day, I had a phone interview with Pearson for a potential part-time job; that interview went well and I’m moving on to the second stage of the employment process. I spent some more time in Spivak’s little book before spending the remainder of my evening working problems from Volume One of Spivak’s magnum opus. Those problems are also “elementary” but they’re a bit harder. The challenge was good.

Today is supposed to be algebra day. Because we only recently were in a position to remedy some previously-existing financial woes, however, we spent most of the day split between running errands and spending time out and about with our son. I did take both Eisenbud/Harris and Perrin with me, along with my trusted G2 and Composition Book; very little progress was made, however.

I’m actually about to dip out for the evening here in a few minutes, but depending on how much energy I have tonight, I might buckle in and try to figure out some of this sheaf theory stuff. If I had a fourth Algebraic Geometry Observation published, it would be that transferring between theory and problems which apply said theory is very VERY difficult.

So it’s now creeping into the third (full) week of June. School got out for me during the first (full) week of May. Regardless of how woeful you may consider your abilities in mathematics, I’m sure you can deduce something very clear from these facts:

Summer is about half over.

Generally, that fact in and of itself wouldn’t be too terrible. I mean, big deal: Half the summer’s over, and I’ve been working throughout. How big of a failure can that really be?

In this case, it’s actually a pretty big one.

Despite my having read pretty much nonstop since summer began, I haven’t really made it very far into anything substantial. Compounded onto that is the fact that I’ve had to abandon a handful of reading projects after making what appeared to be pretty not-terrible progress into them because of various hindrances (usually, a lack of requisite background knowledge).

It’s been a pretty frustrating, pretty not successful summer, objectively.

"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos